To determine the number of ways 4 people can sit next to each other in a movie theater row, we need to consider these 4 people as a single unit or block since they are sitting together.
1. **Treat them as a block**: When we treat the 4 people as a single block, we effectively have one block plus the remaining seats in the row. If there are no restrictions on other seats, we can consider this block taking up 4 consecutive spaces. For the sake of this explanation, let’s assume the row has at least 4 seats available for them to sit together.
2. **Arranging the block**: Within this block, the 4 people can sit in any order. The number of arrangements of 4 people is calculated using the factorial of the number of people, which is 4! (4 factorial).
3. **Calculating the factorial**:
– 4! = 4 × 3 × 2 × 1 = 24.
Therefore, the 4 people can be arranged in 24 different ways inside their block.
4. **Choosing positions**: Now we must consider how many possible blocks can fit in the row of seats. If we assume there are total ‘n’ seats in the row, the block (containing the 4 people) can be located in ‘n-4+1’ positions, meaning the block can start at seat 1, 2, …, up to (n-3).
5. **Total arrangements**: Finally, the total number of ways to arrange the 4 people while sitting next to each other in a row with ‘n’ seats is given by the formula, which combines the positioning of the block and the arrangements within the block. Therefore, we need to multiply the number of block positions by the arrangements of the people:
Total Ways = Number of Positions × Arrangements = (n – 4 + 1) × 4! = (n – 3) × 24.
In conclusion, if you have a row of ‘n’ seats, you can use the formula above to find out how many ways 4 people can sit next to each other!